\(25.\left(\dfrac{-1}{5}\right)^2+\dfrac{1}{5}-9.\left(\dfrac{-1}{9}\right)^2+\left(\dfrac{1}{9}\right)^{20}\)
\(=25.\dfrac{1}{25}+\dfrac{1}{5}-9.\dfrac{1}{81}+\left(\dfrac{1}{9}\right)^{20}\)
\(=1+\dfrac{1}{5}-\dfrac{1}{9}+\left(\dfrac{1}{9}\right)^{20}\)
\(=\dfrac{6}{5}+\dfrac{1}{9}\left[\left(-1\right)+\left(\dfrac{1}{9}\right)^{19}\right]\)
\(25\cdot\left(-\dfrac{1}{5}\right)^2+\dfrac{1}{5}-9\cdot\left(-\dfrac{1}{9}\right)^2+\left(\dfrac{1}{9}\right)^{20}\)
\(=25\cdot\dfrac{1}{25}+\dfrac{1}{5}-9\cdot\dfrac{1}{81}+\dfrac{1}{9^{20}}\)
\(=1+\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{9^{20}}\)
\(=\dfrac{49}{45}+\dfrac{1}{9^{20}}\)
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